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Suppose I have two functions, $f(x)$ and $g(x)$.

For any value of $x$, $f(x) = g(x)$.

If I didn't know that $f(x) = g(x)$, then would I necessarily be able to rearrange $f(x)$ into the form $g(x)$?

An example of when you can do this is for $f(x) = 3x+9, g(x) = 3(x+3)$, where you can obviously factor the $x$ out of $f(x)$. However, can you always do this for any two identical functions?

One pair of functions which comes to mind is $f(x) = \sin(x), g(x) = \cos(x-\pi/2)$, which are, of course, identical, but I wouldn't know how to show that they are.

N. F. Taussig
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Jacob Garby
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  • By using right triangles you can show easily that $\sin x=\cos(90-x)$. – DynamoBlaze Jun 24 '18 at 18:47
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    Definitely not in general. See https://math.stackexchange.com/questions/584230/how-do-you-validate-that-two-math-expressions-are-equal/584261# – MJD Jun 24 '18 at 18:48
  • To make a mathematical question out of this, you'd need to specify (1) how the functions are given to you and (2) what sorts of steps are allowed in rearranging a function. For example, if (1) the functions had to be given as polynomials over the real numbers and (2) you're allowed the usual algebraic manipulations as in high-school algebra, then the answer is yes. But for less trivial (presentations of) functions, the answer isn likely to be no. – Andreas Blass Jun 24 '18 at 23:37

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